# General Concepts

The main difference between amplifiers and lasers, as also discussed under the “Lasers” portion of this website, is that amplifiers are essentially just lasers without feedback. The operating wavelength, gain bandwidth, and relevant rate equations are all determined by the doping elements rather than the host medium. The doping elements are in fact what provide the gain, as the silica fiber by itself would just add to the dispersion and nonlinear effects without amplifying the signal. Like in a laser, the pumping scheme is usually determined by the resonant absorption peak at a given wavelength, and then the gain of the amplifier results through the interaction of the signal with the excited system.

GAIN COEFFICIENT

The gain coefficient of a homogeneously broadened gain medium is written as:

$$g(\omega) = \frac{g_0}{1+(\omega-\omega_a)^2T_2^2+\frac{P}{P_s}}$$

where “$g_0$” is the peak value of the gain which is determined by the dopant of the amplifier, “$\omega$” is the frequency of the incident signal, “$\omega_a$” is the atomic transition frequency, “$P$” is the power of the signal being amplified, “$P_s$” is the saturation power, and “$T_2$” is the dipole relaxation time. As discussed also in the “Lasers” portion, the saturation power “$P_s$” is determined by the dopant parameters such as the population fluorescence time $T_1$ and the transition cross section $\sigma$. This is the large signal gain coefficient, because as can be seen from the equation, the total gain is decreased by an increasing input signal power $> P_s$. Likewise, when $P << P_s$ such that $\frac{P}{P_s} \approx 0$, then the gain coefficient is maximized. The condition where the fractional power term equals 0 is the small signal condition, and is written more explicitly below. Also, note that there is no time dependence in this definition of the gain coefficient; the time dependent case is given in the “Ultrashort Pulse Amplification” area of this subsection here.

AMPLIFIER GAIN AND BANDWIDTH

The small signal gain coefficient is given as

$$g(\omega) = \frac{g_0}{1+(\omega-\omega_a)^2T_2^2}$$

where, as we noted above, the power dependence has vanished and the gain only depends on material parameters as well as the signal frequency. Note from this equation that as the frequency of the signal $\omega$ is further detuned from the atomic resonance frequency $\omega_a$, the gain is decreased, and is maximized when $\omega \approx \omega_a$. With the assumption that the gain profile is homogeneously broadened, the gain is governed by a Lorentzian lineshape. This can be seen in the figure below.

The gain bandwidth is given as:

$$\delta{v_g} = \frac{\delta\omega_g}{2\pi} = \frac{1}{\pi{T_2}}$$

It is interesting to note here that the gain bandwidth is dependent on the dipole relaxation time. The effect of strong homogeneous or homogeneous broadening then means a shorter dipole relaxation time, which of course means a larger gain bandwidth. We can see the physical reasons for this much more easily in bulk solid state and gas lasers, so the majority of the physical discussion of the broadening is in this section of the “Lasers” portion of this website.

A related concept to the gain bandwidth is the amplifier bandwidth. Often these two terms are used interchangeably, even though their formulation is slightly different as we shall see shortly. The amplification factor of the fiber amplifier is given as:

$$G = \frac{P_{out}}{P_in}$$

where the value of $P_{in}$ is determined by the power of the cw signal being amplified, and $P_{out}$ is determined by the amplifier itself.

The gain or amplification bandwidth is important especially in ultrashort pulse amplification, because it is the finite bandwidth that limits the pulse duration of ultrashort pulses. As a signal with a bandwidth greater than that of the amplifier or gain spectra is amplified, only the regions of the signal spectrum that correspond to the amplifier spectrum will be amplified. The spectrum of the ultrashort pulse is then shortened, such that on output from the amplifier the time-bandwidth product is limited to longer pulse duration. This is called gain-narrowing, and poses a fundamental limit presently in ultrashort pulse, high peak power applications of fiber amplifiers.

GAIN SATURATION

AMPLIFIER NOISE

AMPLIFIER APPLICATIONS